Solving Quadratic Equations for IGCSE Mathematics | IGCSE 数学：二次方程求解

📚 Solving Quadratic Equations for IGCSE Mathematics | IGCSE 数学：二次方程求解

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and graphs to real-life modelling. This revision guide explains the standard form, the main solving methods, the discriminant, graphical interpretation, and common exam pitfalls.

二次方程贯穿 IGCSE 数学大纲，从代数、图像到实际问题建模都会出现。本复习指南将讲解标准形式、主要解法、判别式、图像意义以及考试中常见的失分点。

1. What Is a Quadratic Equation? | 什么是二次方程

A quadratic equation is a polynomial equation of degree 2. The highest power of the variable is x².

二次方程是一个次数为 2 的多项式方程，其变量的最高次幂是 x²。

The general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

一般形式为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。

If a = 0, the equation becomes linear, not quadratic. For example, x² − 3x + 2 = 0 is quadratic, but 2x + 1 = 0 is linear.

如果 a = 0，方程就变成一次方程，而不是二次方程。例如，x² − 3x + 2 = 0 是二次方程，而 2x + 1 = 0 是一次方程。

2. Standard Form and Key Terms | 标准形式与关键术语

In ax² + bx + c = 0, the term ax² is the quadratic term, bx is the linear term, and c is the constant term. The coefficient a is called the leading coefficient.

在 ax² + bx + c = 0 中，ax² 是二次项，bx 是一次项，c 是常数项。系数 a 称为首项系数。

A root or solution is a value of x that makes the equation true. Quadratic equations can have two real roots, one repeated real root, or no real roots.

根或解是使方程成立的 x 值。二次方程可以有两个实数根、一个重复实数根，或没有实数根。

ax² + bx + c = 0, a ≠ 0

Always rewrite an equation into this standard form before choosing a solution method.

在选择题型解法之前，务必先将方程整理成这个标准形式。

3. Solving by Factorisation | 因式分解法

Factorisation is usually the fastest method when the quadratic expression factorises neatly. Follow these steps: set one side to 0, factorise, set each bracket equal to 0, and solve.

当二次式可以容易地因式分解时，因式分解法通常是最快的方法。步骤如下：将一边写成 0，进行因式分解，令每个括号等于 0，然后解方程。

Example 1: Solve x² + 7x + 12 = 0.

例 1：解 x² + 7x + 12 = 0。

x² + 7x + 12 = (x + 3)(x + 4) = 0

Therefore x + 3 = 0 or x + 4 = 0, giving x = −3 or x = −4.

因此 x + 3 = 0 或 x + 4 = 0，得到 x = −3 或 x = −4。

Example 2: Solve 2x² + 5x − 3 = 0.

例 2：解 2x² + 5x − 3 = 0

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